Rational map $ax+1\mathrm{/}x$ on the projective line over $\mathbb{Q}\mathrm{\_}p$

The dynamical structure of the rational map $ax+1\mathrm{/}x$ on the projective line $\mathrm{\P }$ over the field $\mathbb{Q}\mathrm{\_}p$ of $p$-adic numbers is described for $p\geq 3$.